Midras Mazmur 51:19: Ein Yaakov (Glick Edition), Vayikra Rabbah, Mekhilta DeRabbi Yishmael & Bamidbar Rabbah

Hezekiah said: "A man's prayer is not heard unless he considers himself as soft-hearted as flesh, [but not hard like a rock]; for it is said (Is. 66, 23) And it shall come to pass that from one new moon to the other new moon, and that from one Sabbath to the other Sabbath, should all flesh come to prositorious act in this world, it proceeds him in trate themselves before Me, Saith the Lord." R. Zera said: "Concerning flesh it is written (Lev. 13, 18) It is healed but concerning man the word 'healed' is not written." R. Jochanan said: "The word Adam means Aphar (dust), daam blood, and mass gall; Bossor (flesh) means busha (disgrace), scrucha (smell) and Zima (worms). According to others it means: Busha (disgrace) (grace), and Rima (worms), for it for it is written with a Shin. R. Ashi said: "Everyman who possesses haughtiness will wane in reputation, as it is said (Ib. 14, 56) And for a swelling for a rising and the word S'eth (rising) refers to haughtiness, as it is said (Is. 2, 14) And over all the high mountains (nissa'ah) and over all the exalted hills and sapachas (swelling) refers to nothing else but an ill-considered thing, as it is said (I Sam. 2, 36) And he will say, 'Attach me, (Safcheni) I pray thee, unto one of the priestly officers that I may eat all bread.'" R. Joshua b. Levi said: "Come and see how greatly are the humble-spirited considered before the Holy One, praised be He, for at the time the Temple was in existence if a man brought a burnt-offering sacrifice, he received the reward of a burnt-offering, and if one brought a meal-offering he received the reward of a meal-offering, but a man who who possesses a humble spirit is considered by Scripture as if he had brought all kinds of sacrifices, as it is said (Ps. 51, 19) The sacrifice of God is a broken spirit. And not only this but even his prayer is not rejected, as it is said further A broken and contrite heart, O God wilt Thou not despise."

“Offerings to God are a broken spirit…” (Psalms 51:19). Zavdi ben Levi, Rabbi Yosei ben Petras, and the Rabbis, one said: David said before the Holy One blessed be He: ‘I overcame my inclination and repented before You. If You accept me in repentance, I know that Solomon my son will stand and build the Temple, build the altar, and burn all the offerings in the Torah upon it,’ based on this verse: “Offerings to God are a broken spirit.” Another said: From where is it derived that one who repents, it is ascribed to him as though he ascended to Jerusalem, built the Temple, built the altar, and sacrificed all the offerings in the Torah upon it? It is from this verse: “Offerings to God are a broken spirit.” The Rabbis say: From where is it derived that one who passes before the ark⁷One who serves as a prayer leader. must mention the Temple service and the offerings, and bow [when he concludes] this blessing: Find favor, our God, rest in Zion, may Your children worship You?⁸This is a variation of the blessing “Find favor [retzei]” recited today in the Amida prayer. There are some who sought to derive it from this: “Offerings to God are a broken spirit.” Rabbi Abba bar Yudan said: Everything that the Holy One blessed be He deemed unfit in an animal He deemed fit in a person. He deemed unfit in an animal: “Blind, broken, maimed, or with a wart” (Leviticus 22:22), but He deemed fit in a person: “A broken and crushed heart” (Psalms 51:19).
Rabbi Alexandri said: The layman, if he uses broken utensils, it is demeaning for him. But the vessels of the Holy One blessed be He are broken, as it is stated: “The Lord is near to the brokenhearted” (Psalms 34:19); “He heals the brokenhearted” (Psalms 147:3); “[I dwell in the high and holy place, with] the downtrodden and the humble” (Isaiah 57:15); “Offerings to God are a broken spirit.”
“A broken…heart,” Rabbi Abba bar Yudan in the name of Rabbi Yuda bar Rabbi Simon: This is analogous to a king who was walking in the wilderness and his friend came and gave him a tribute of one basket of figs and one barrel of wine. He said to him: ‘Is this a great tribute?’ He said to him: ‘My lord, the king, I have given you a provisional tribute, but when you enter your palace, you will see how much tribute I will give you.’ So too, the Holy One blessed be He said to Israel: “This is the law of the burnt offering: It is the burnt offering” (Leviticus 6:2). They said before Him: ‘Master of the universe, we have sacrificed before You provisionally, but when You show favor “with Your will to Zion; build the walls of Jerusalem, then You will delight in righteous sacrifices, in burnt offerings and whole burnt offerings”’ (Psalms 51:20–21).

(Ibid.) "And Moses entered into the mist": This (his closeness to the L–rd) was a function of his humility, viz. (Numbers 12:3) "And the man Moses was extremely humble, etc." Scripture hereby apprises us that if one is (truly) humble, in the end, he will cause the Shechinah to repose upon a man upon the earth, viz. (Isaiah 57:15) "For thus said the High and Exalted, who abides forever and whose name is holy (… I dwell with the oppressed and with the lowly of spirit, etc."), and (Ibid. 61:1) "The spirit of the L–rd G–d is upon me, for the L–rd has anointed me to bring tidings to the humble, etc.", and (Ibid. 66;2) "And all of these things My hand created (… but it is to this that I look, to the poor and broken-spirited, etc."), and (Psalms 51:19) "The sacrifices of G–d are a broken spirit. A heart broken and oppressed, G–d will not despise."